Finite-lattice form factors in free-fermion models

TL;DR

This paper derives exact finite-lattice form factors for a broad class of free-fermion models, including the Ising model, by translating Kaufman's fermionic approach into Grassmann integrals, revealing eigenvector correspondences and enabling precise calculations.

Contribution

It introduces a method to compute finite-lattice form factors in free-fermion models using Grassmann integrals, unifying various models under a common framework.

Findings

Eigenvectors of transfer matrices are explicitly determined.

Exact finite-lattice form factors of spin operators are obtained.

The approach applies to models including the Ising and XY chains.

Abstract

We consider the general $\mathbb{Z}_2$-symmetric free-fermion model on the finite periodic lattice, which includes as special cases the Ising model on the square and triangular lattices and $\mathbb{Z}_n$-symmetric BBS $\tau^{(2)}$-model with $n=2$. Translating Kaufman's fermionic approach to diagonalization of Ising-like transfer matrices into the language of Grassmann integrals, we determine the transfer matrix eigenvectors and observe that they coincide with the eigenvectors of a square lattice Ising transfer matrix. This allows to find exact finite-lattice form factors of spin operators for the statistical model and the associated finite-length quantum chains, of which the most general is equivalent to the XY chain in a transverse field.